 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/m/2/c/i/solution

After the first step, a chain starting at one enters the closed class

{2, 3}

p

, starting at two, or enters

{4, 5}

1 - p

and can never reach three. Within the first class, the probability of going from its first state to its second in

n - 1

steps is the complement of the formula in part (b). Thus

P (X_{n} = 3 ∣ X_{0} = 1) = p \frac{a}{a + b} [1 - (1 - a - b)^{n - 1}], n \geq 1.

n = 1

this correctly gives zero.

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